Inspired by the work of Ringel,
Lusztig ([1]}, [2]}) gave a categorification of \(\mathbf {U}_{\nu }^-\)  by using an additive subcategory \(\mathcal {Q}_{\nu }\)  of \({\mathcal {D}}_{G_{\mathbf {V}}}^{b,ss}(E_{\mathbf {V}})\) .
The multiplication and comultiplication in \(\mathbf {U}^-\)  are induced by induction functors \(\mathrm {Ind}^{\alpha +\beta }_{\alpha ,\beta }\) 
and restriction functors \(\mathrm {Res}^{\alpha +\beta }_{\alpha ,\beta }\) , respectively.
